Having discovered the model of planets traveling on elliptical orbits, Kepler faced the problem of finding a simple geometrical method of deducing the elements of such orbits, which include the eccentricity e and the direction of the line of apsides ψ, from observations. Instead, he could only obtain these values by trial and error. Once they have been found, Kepler wanted to determine geometrically from his area law the position of planets. This is the focus of discussion on pages 114 and 115 of Heilbron’s book which shall be elaborated on here, with reference to W.M.Smart’s “Textbook in Spherical Astronomy”.
Let the radius SZ make the angle θ with SP; θ is called the true anomaly. Let a circle be described on the major axis AP as diameter; its radius is thus a. Let RZ, the perpendicular from Z to AP, be produced to meet this circle at Q. Then angle QCP is called the eccentric anomaly, denoted by η. By properties of ellipses,
RZ : RQ = b : a (1)
where a is the semi-major axis CP.
Since RZ = rsinθ and RQ = CQ sin η = a sin η then by (1),
Since SR = r cos θ and also, SR = CR - CS = and therefore
(2)2 +(3)2 gives
Using the relation and after a little reduction, we obtain
Since , we have . Upon substituting terms on the right hand side of the equation with (3) and (4) and manipulation, we obtain
Divide (5) by (6), and taking the square root, we get
Equations (4) and (7) therefore express the radius vector r and the true anomaly θ in terms of the eccentric anomaly η. To obtain η, we turn towards solving Kepler’s Equation.
Recall that w denotes mean angular velocity and the product w(t-τ) represents the angle described in an interval (t-τ) by a radius vector rotating about S with constant angular velocity w. We define the mean anomaly, denoted by M, such that
M = w(t-τ).
In Figure 30, the area SZP is thus given by
To express the area in terms of the eccentric anomaly η, we consider area SZP as the sum of area ZSR and area RZP. Take first the area of triangle ZSR. Its area is .
area ZSR =
Next consider area RZP. By application of Cavalieri’s Principle to ellipses, we know that area RZP is equal to times of area QRP. But area QRP is area of sector CQP minus the area of triangle QCR, where angle QCR is η, area CQP is and area QCR is or Hence,
Then adding areas ZSR and RZP gives
Using the two equations for area SZP, we then have
This is Kepler’s Equation which relates the eccentric anomaly η and the mean anomaly M. If M and e are known, it is then possible to determine the corresponding value of η. Thereafter, it is possible to find the value of the true anomaly θ by substituting the values of the eccentricity e and found value of eccentric anomaly η into the earlier equation for θ in terms of η; this then allows us to determine the position of planet Z at time t.
Nonetheless, just as indicated in Heilbron’s book, even though the Kepler’s Equation is simple to write down, it can be solved only by guesswork and successive approximations. I shall not attempt to demonstrate how this is done exactly but a good reference for the detailed steps is found in Smart’s book, pages 117 – 119.
Seth Ward, professor of geometry at Oxford, had thought he found a geometrical method that could solve Kepler’s problem simply and accurately.
With reference to Figure 31, AP is the line of apsides, E and S are Earth and Sun respectively, X marks the equant point in the unoccupied focus of the ellipse, and XS = ae. Let the mean anomaly be M whilst the true anomaly be θ’. By extending XE to Q such that EQ = SE, and by a basic property of ellipses, we have XQ = AP = 2a. In triangle XSQ, by Sine Rule,
Since we have
Using the above expression, the following can be deduced,
This equation relates the true anomaly to the apsidal distances and mean anomaly. Hence, Ward thought he had managed to devise a geometrical method that is simple to apply and would give the value of the true anomaly directly. Unfortunately, Ward was wrong; his method did not give the exact value of the true anomaly. The following gives an explanation of why this occurred, with reference to Appendix E.
According to Figure 32, the true anomaly is θ = η - α + β.
In triangle CPR,
By Cavalieri’s Principle,
However in triangle CRQ,
By substitution, we can get
where a is small.
By properties of ellipses,
In addition, a is small. Thus,
Similarly, in triangle CPF, by Sine Rule,
b is small and thus by approximation,
To express sin η in terms of e and M, we have
To express cosh in terms of e and M, we have
Referring to Figure 27, in triangle XSQ, by Sine Rule,
Consider a a small angle. Then by small angle approximations,
Since we have
Thus, In other words, there was a discrepancy of a maximum between Ward’s found value for “true anomaly” and the actual true anomaly.